The Rangers grabbed a 3-2 lead in the best-of-seven series and will look to advance when they host Game 6 on Saturday.
Mats Zuccarello wasted a glittering chance to give New York an early lead when he couldn't lift the puck over Price's pad from the doorstep in the opening minute.
Montreal then struck first when Lehkonen fought off Marc Staal to put a spin-around shot off the side of the net and score on a wraparound at 12:07.
The Rangers responded with a short-handed goal. Kevin Hayes was in the box for holding when Fast finished a 2-on-2 counterattack after taking a slick feed from Zibanejad at 15:56.
Only 25 seconds later, Andrei Markov slipped a pass into the slot for an unguarded Gallagher to score on a snap shot.
Skjei tied it at 18:28 of the second frame. Rick Nash took the puck to the net and Skjei knocked the rebound into an open side.
Skjei became the first Rangers rookie defenseman with two playoff goals since Brian Leetch had three in 1989.
Montreal forward Phillip Danault hit a post on a short-handed chance in the opening minute and Max Pacioretty was stopped on a breakaway by Lundqvist in a scoreless third period in which New York had most of the sustained pressure.
Kreider had a chance to end the game 10 minutes into overtime but whiffed on a Zibanejad pass at the doorstep.
NOTES: Alexei Emelin saw his first action of the series on Montreal's third defense pairing, replacing Brandon Davidson. The Russian missed six games with an undisclosed injury.
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